How to extrude rounded and smooth?
You can't apply a subsurf to part of an object, but you can control the way it subdivides.  Consider this object



If I apply the subsurf I get



It solves your problem by rounding, but it rounds where you don't want it to. There are several ways to fix this, depending on your need.  Here are a couple.

You can bevel edges and control how the rounding happens, perhaps making it less obvious


Select edges you want to control

Enter CTRL-B. The cursor will change.  Moving it controls the width of the bevel. Rotating the scroll wheel controls the number of steps


You can use the bevel modifier to accomplish the same thing but with more control.

You can create edge loops.  Consider the base of the extrusions. Enter CTRL-R. Place your cursor near the base of it until a red line appears around the extrusion


Left-Click and then move the line where you want it

Left-Click again.

My example would require a lot of edge loops or a lot of bevels. Here's a version with all of the edge loops selected.